Question: Factor the following expression: $5$ $x^2+$ $4$ $x$ $-9$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(-9)} &=& -45 \\ {a} + {b} &=& & & {4} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-45$ and add them together. Remember, since $-45$ is negative, one of the factors must be negative. The factors that add up to ${4}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({9})({-5}) &=& -45 \\ {a} + {b} &=& {9} + {-5} &=& 4 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 +{9}x {-5}x {-9} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 +{9}x) + ({-5}x {-9}) $ Factor out the common factors: $ x(5x + 9) - 1(5x + 9) $ Notice how $(5x + 9)$ has become a common factor. Factor this out to find the answer. $(5x + 9)(x - 1)$